2005/10/12 by Ramacher, Pablo
#20C99 #35P20 #47G30 #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.math/0510261
Let G⊂ Ø(n) be a group of isometries acting on n-dimensional Euclidean space \Rn, and \bfX a bounded domain in \Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A0 in Ł2(\Rn) with G-invariant Weyl symbol, and assume that it is semi-bounded from below. We show that the spectrum of the Friedrichs extension A of the operator res ∘ A0 ∘ ext: \CT(\bfX) → Ł2(\bfX) is discrete, and derive asymptotics for the number Nχ(λ) of eigenvalues of A less or equal λ and with eigenfunctions in the χ-isotypic component of Ł2(\bfX), giving also an estimate for the remainder term in both cases where G is a finite, or, more generally, a compact group. In particular, we show that the multiplicity of each unitary irreducible representation in Ł2(\bfX) is asymptotically proportional to its dimension.